Mechanical Characterization of Materials and Wave Dispersion E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 461

Veröffentlichungsjahr: 2013

Beschreibung

Over the last 50 years, the methods of investigating dynamic properties have resulted in significant advances. This book explores dynamic testing, the methods used, and the experiments performed, placing a particular emphasis on the context of bounded medium elastodynamics. Dynamic tests have proven to be as efficient as static tests and are often easier to use at lower frequency. The discussion is divided into four parts. Part A focuses on the complements of continuum mechanics. Part B concerns the various types of rod vibrations: extensional, bending, and torsional. Part C is devoted to mechanical and electronic instrumentation, and guidelines for which experimental set-up should be used are given. Part D concentrates on experiments and experimental interpretations of elastic or viscolelastic moduli. In addition, several chapters contain practical examples alongside theoretical discussion to facilitate the readers understanding. The results presented are the culmination of over 30 years of research by the authors and as such will be of great interest to anyone involved in this field.

Leseprobe

Table of Contents

Preface

Acknowledgements

PART I Mechanical and Electronic Instrumentation

Chapter 1: Guidelines for Choosing the Experimental Set-up

1.1. Choice of matrix coefficient to be evaluated and type of wave to be adopted

1.2. Influence of frequency range

1.3. Dimensions and shape of the samples

1.4. Tests at high and low temperature

1.5. Sample holder at high temperature

1.6. Visual observation inside the ambient room

1.7. Complex moduli of viscoelastic materials and damping capacity measurements

1.8. Previsional calculation of composite materials

1.9. Bibliography

Chapter 2: Review of Industrial Analyzers for Material Characterization

2.1. Rheovibron and its successive versions

2.2. Dynamic mechanical analyzer DMA 01dB – Metravib and VHF 104 Metravib analyzer

2.3. Bruel and Kjaer complex modulus apparatus (Oberst Apparatus)

2.4. Dynamic mechanical analyzer DMA–Dupont de Nemours 980

2.5. Elasticimeter using progressive wave PPM 5

2.6. Bibliography

Chapter 3: Mechanical Part of the Vibration Test Bench

3.1. Clamping end

3.2. Length correction

3.3. Supported end

3.4. Additional weight or additional torsion lever used as a boundary condition

3.5. Free end

3.6. Pseudo-clamping sample attachment

3.7. Sample suspended by taut threads

3.8. Sample on foam rubber plate serving as a mattress

3.9. Climatic chamber

3.10. Vacuum system

3.11. Bibliography

Chapter 4: Exciters and Excitation Signals

4.1. Frequency ranges

4.2. Power

4.3. Nature and performance of various exciters

4.4. Room required for exciter installation

4.5. Details for electrodynamic shakers

4.6. Low cost electromagnetic exciters with permanent magnet

4.7. Piezoelectric and ferroelectric exciters

4.8. Design of special ferroelectric transducers

4.9. Power piezoelectric exciters

4.10. Technical details concerning ultrasonic emitters for the measurement of material stiffness coefficients on ultrasonic test benches

4.11. Bibliography

4.12. Appendix 4A. Example of ferroelectric plates and disks

Chapter 5: Transducers

5.1. Introduction

5.2. Transducers and their principal performance

5.3. The main classes of fixed reference transducers

5.4. Condenser-type transducer

5.5. Inductance transducers

5.6. Mutual inductance transducer

5.7. Differential transformer transducer

5.8. Contactless inductance transducer with a permanent magnet

5.9. Eddy current transducer

5.10. Seismic transducers

5.11. Piezoresistive accelerometer

5.12. Other transducers

5.13. Force transducers

5.14. Bibliography

5.15. Appendix 5A. Condenser with polarization

5.16. Appendix 5B. Eigenfrequencies of some force transducers: Rayleigh and Rayleigh-Ritz upper bound methods

5B.1. Rayleigh's method

5B.2. Rayleigh-Ritz's method

5B.3. Preliminary experimental test on the force transducer

Chapter 6: Electronic Instrumentation, Connecting Precautions and Signal Processing

6.1. Preamplifiers and signal conditioners following the transducers

6.2. Cables and wiring considerations

6.3. Transducer selection and mountings

6.4. Transducer calibration

6.5. Digital signal processing systems: an overview

6.6. Other signal processing programs

6.7. Reasoned choice of excitation signals

6.8. Bibliography

6.9. Appendix 6A. The Shannon theorem and aliasing phenomenon

6.10. Appendix 6B. Time window (or weighting function)

6B.1. Kaiser-Bessel window

6B.2. Hamming window

Chapter 7: The Frequency Hilbert Transform and Detection of Hidden Non-linearities in Frequency Responses

7.1. Introduction

7.2. Mathematical expression of the Hilbert transform

7.3. Kramer-Kronig's relationships

7.4. Causal signal and Fourier transform

7.5. Hilbert transform of a truncated transfer function

7.6. Impulse response of a system. Non-causality due to measurement defects

7.7. Summary of principal result in sections 7.5 and 7.6

7.8. Causalized Hilbert transform

7.9. Some practical aspects of Hilbert transform computation

7.10. Conclusion

7.11. Bibliography

7.12. Appendix 7A. Line integral of complex function and Cauchy's integral

7A.1. Analyticity of a function f(z) of complex variable z

7A.2. Expression of Cauchy's integral of the function f(z)/(z-α)

7.13. Appendix 7B. Hilbert transform obtained directly by Guillemin's method

Chapter 8: Measurement of Structural Damping

8.1. Introduction

8.2. Overview of various methods used to evaluate damping ratios in structural dynamics

8.3. Measurement of structural damping coefficient by multimodal analysis

8.4. The Hilbert envelope time domain method

8.5. Detection of hidden non-linearities

8.6. How to relate material damping to structural damping?

8.7. Concluding remarks

8.8. Bibliography

PART II Realization of Experimental Set-ups and Interpretation of Measurements

Chapter 9: Torsion Test Benches: Instrumentation and Experimental Results

9.1. Introduction

9.2. Industrial torsion test bench

9.3. Parasitic bending vibration of rod

9.4. Shear moduli of transverse isotropic materials

9.5. Elastic moduli obtained for various materials

9.6. Experimental set-up to obtain dispersion curves in a large frequency range

9.7. Experimental results obtained on short samples

9.8. Experimental wave dispersion curves obtained by torsional vibrations of a rod with rectangular cross-section

9.9. Frequency spectrum for isotropic metallic materials (aluminum and steel alloy)

9.10. Impact test on viscoelastic high damping material

9.11. Concluding remarks

9.12. Bibliography

9.13. Appendix 9A. Choice of equations of motion

9A.1. Circular cross-section

9A.2. Square cross-section

9A.3. Rectangular cross-section

9A.4. Ratio of Young's modulus to shear modulus

9A.5. Special experimental studies of wave dispersion phenomenon

9.14. Appendix 9B. Complementary information concerning formulae used to interprete torsion tests

9B.1. Quick overview of Saint Venant's theory applied to the problem of dynamic torsion

9.15. Appendix 9C: details concerning the βT(c) function in the calculation of rod stiffness CT

9.16. Appendix 9D. Compliments concerning the solution of equations of motion with first order theory

9D.1. Displacement field

9D.2. Relations between two sets of coefficients

9D.3 Equations giving the two sets of coefficients Aa,Ba,Ca,Da deduced from the four boundary conditions

9D.4. Evaluation of coefficients in [9.D.6]

9D.5. Equations in Aa , Ba , Ca , Da deduced from the four boundary conditions

Chapter 10: Bending Vibration of Rod Instrumentation and Measurements

10.1. Introduction

10.2. Realization of an elasticimeter

10.3. How to conduct bending tests

10.4. Concluding remarks

10.5 Bibliography

10.6. Appendix 10A. Useful formulae to evaluate the Young's modulus by bending vibration of rods

10A.1. Bernoulli-Euler's equation

10A.2. Timoshenko-Mindlin's equation

10A.3. Boundary conditions and wave number equation

10A.4. Important parameters in rod bending vibration

10A.5. Expression of wave number

10A.6. Young's modulus (Bernoulli's theory)

10A.7. Young's modulus (Timoshenko-Mindlin's equation)

Chapter 11: Longitudinal Vibration of Rods: Material Characterization and Experimental Dispersion Curves

11.1. Introduction

11.2. Mechanical set-up

11.3. Electronic set-up

11.4. Estimation of phase velocity

11.5. Short samples and eigenvalue calculations for various materials

11.6. Experimental results interpreted by the two theories

11.8. Experimental results obtained with short rod

11.9. Concluding remarks

11.10. Bibliography

11.11. Appendix 11A. Eigenvalue equation for rod of finite length

11.12. Appendix 11B. Additional information concerning solutions of Touratier's equations

11B.1. Eigenequation with elementary theory of motion

Chapter 12: Realization of Le Rolland-Sorin's Double Pendulum and Some Experimental Results

12.1. Introduction

12.2. Principal mechanical parts of the double pendulum system

12.3. Instrumentation

12.4. Experimental precautions

12.5. Details and characteristics of the elasticimeter

12.6. Some experimental results

12.7. Damping ratio estimation by logarithmic decrement method

12.8. Concluding remarks

12.9. Bibliography

12.10. Appendix 12A. Equations of motion for the set (pendulums, platform and sample) and Young's modulus calculation deduced from bending tests

12A.1. Equations of motion

12A.2. Solutions for pendulum oscillations

12A.3. Relationship between beating period τ and sample stiffness k

12A.4. Young's modulus calculation

12.11. Appendix 12B. Evaluation of shear modulus by torsion tests

12B.1. Energy expression

Chapter 13: Stationary and Progressive Waves in Rings and Hollow Cylinders

13.1. Introduction

13.2. Choosing the samples based on material symmetry

13.3. Practical realization of a special elasticimeter for curved beams and rings: in plane bending vibrations

13.4. Ultrasonic benches

13.5. Experimental results and interpretation

13.6. List of symbols

13.7. Bibliography

13.8. Appendix 13A. Evaluation of Young's modulus by using in plane bending motion of the ring

13.9. Appendix 13B. Determination of inertia moment of a solid by means of a three-string pendulum

13B.1. Principle of the method

13B.2. Calculations

13.10. Appendix 13C. Necessary formulae to evaluate Young's modulus of a straight beam

Chapter 14: Ultrasonic Benches: Characterization of Materials by Wave Propagation Techniques

14.1. Introduction

14.2. Ultrasonic transducers

14.3. Pulse generator

14.4. Mechanical realization of ultrasonic benches

14.5. Experimental interpretation of phase velocity and group velocity

14.6. Some experimental results on composite materials

14.7. Viscoelastic characterization of materials by ultrasonic waves

14.8. Bibliography

14.9. Appendix 14A. Oblique incidence and energy propagation direction [KLI 92]

14.10. Appendix 14.B. Water immersion bench, measurement of coefficients of stiffness matrix

14B.1. Expression of phase velocity in the sample

14B.2. Phase velocity measurement by propagation time (δt) evaluation

14B.3. Phase velocity evaluation without time measurements

Chapter 15: Wave Dispersion in Rods with a Rectangular Cross-section: Higher Order Theory and Experimentation

15.1. Introduction

15.2. Summary table of some wave dispersion research

15.3. Longitudinal wave dispersion: influence of the material and geometry of the bounded medium

15.4. Bending wave dispersion

15.5. First order for torsional motion in a transverse isotropic rod

15.6. Interest in theories with higher degrees of approximation

15.7. Experimental set-ups to visualize stationary waves in rods

15.8. Electronic set-up and observed signals on a multi-channel oscilloscope

15.9. Presentation of experimental results

15.10. Concluding remarks

15.11. Bibliography

15.12. Appendix 15A. Touratier's theory using Hellinger-Reissner's mixed fields

15A.1. Outline of Touratier's mixed field theory

15A.2. General equations deduced from the two fields principle

15A.3. Formulation of the boundary condition problem

15A.4. Symmetry considerations concerning the three kinds of motion

15A.5. Truncating process for one dimensional theories: extensional waves

15A.6. Equations of motion for extensional movement

15A.7. Effective front velocity and wave front velocity

15A.8. Bending equations of motion

15A.9. Equations of motion: torsional vibration

15.13. Appendix 15B. Third order Touratier's theory

15B.1. Extensional waves with nine evaluated modes

15B.2. Geometrical characteristics of displacement components ujmn and physical interpretation

15B.3. Bending mode in the direction - geometrical interpretation

15B.4. Shear motion around longitudinal rod axis

List of Authors

Index

First published 2010 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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© ISTE Ltd 2010

The rights of Yvon Chevalier and Jean Tuong Vinh to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Cataloging-in-Publication Data

Mechanical characterization of materials and wave dispersion : instrumentation and experiment interpretation / edited by Yvon Chevalier, Jean Vinh Tuong.     p. cm.   Includes bibliographical references and index.   ISBN 978-1-84821-193-3  1. Materials--Mechanical properties--Experiments. 2. Structural engineering--Materials--Experiments. 3. Wave motion, Theory of--Experiments. 4. Dispersion--Experiments. 5. Engineering instruments. I. Chevalier, Yvon. II. Vinh, Jean Tuong.  TA404.8.M436 2010  620.1'1292--dc22

2010015277

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-84821-193-3

Preface

In the world of the mechanical characterization of materials, the activities of researchers and engineers can generally be classified into three main areas:

1. designing and building appropriate instruments, covering both mechanical and electronic aspects;

2. conducting experiments to obtain dynamic responses to the bounded medium constituted by the sample, using signal processing to obtain appropriate dynamic responses from the sample;

3. searching for a solution to the inverse problem in viscoelasticity. The dynamics of the sample being known, the dynamic characteristics of the material are sought so as to obtain a material response, either in the field of frequency or time.

We will examine each of these steps in turn below, enabling us to present our point of view before presenting commentaries concerning each chapter of the book.

The conception and realization of appropriate instruments

At first glance, this heading might surprise some readers, given the number of commercially available instruments in applied viscoelasticity. However, in spite of the growing number of these instruments, rarely do they cover all the needs of researchers in this domain. Most of this apparatus does not cover a large frequency range. Often, the physical phenomenon of geometric wave dispersion in the bounded medium constituted by the sample itself is ignored. Evaluation of the material's viscoelastic parameters becomes problematic and might be subject to major error.

Discussion concerning adopted practical boundary conditions is often absent and the conception of the sample holder is such that parasitic coupling between vibrations of different kinds (bending, torsional, longitudinal) occur in the sample and empirical correcting terms are proposed.

In many cases, researchers have to conceive and make the apparatus themselves, to cover a very wide range frequency range from just a few Hertz up to or beyond 100,000 Hz, in which cases special transducers are especially fabricated from ferroelectric plates.

In some situations, a large range of negative and positive temperatures are used in the experiments. In these cases, the judicious choice of boundary conditions is important, taking into account the possible creep of the sample at high temperature.

Electronic instrumentation testing is facilitated by the profusion of apparatus in the field of dynamic testing of structures. With some precautions, experimenters can use them, the special use being essentially different from that frequently adopted in the dynamics of structures. The size of the sample in mechanical characterization is often much smaller than the usual size of mechanical structures compared to the size of the usual transducers themselves.

Conducting experiments on a sample in view of the dynamic characterization of the material

There are four steps in this stage: choice of boundary conditions, search for solution of direct problem, signal processing, and, finally, the search for a solution to the inverse problem. Items one and three will be explained as follows:

The choice of sample boundary conditions

This important choice depends on a variety of parameters which govern the dynamic performances of the sample: frequency range, transducer inertias, eventual extreme ambient temperatures, nature of the vibration imposed on the sample and the sample environment.

Signal processing

This concerns a set of tests which permit the detection of hidden non-linearities in sample responses which are due to transducer responses, the excitation of signal amplitudes. Recent advances in the detection of hidden non-linearities by using the Hilbert frequency transform deserve the experimenter's attention.

Modal analysis in the domain of structural dynamics offers a variety of computer programs available for treating either the transient time response of the structure or its frequency response.

Solving specific viscoelastic problems

There are three steps in this final stage:

– mathematical modeling of the viscoelastic sample, the nature of the vibration being known;

– research into a solution for the material viscoelastic parameters of the sample in the framework of an inverse problem, the dynamic response of the sample being known;

– research into a closed form expression of a viscoelastic modulus versus the frequency or the corresponding relaxation or creep function versus time.

Mathematical modeling of a sample

This concerns a specific problem in which a set of sample elastic coefficients are known, and in which the dynamic responses are deduced from an appropriate theory chosen by the experimenter themselves (see [CHE 10]). This step is important so as to include all the dynamic and static effects which occur in the sample. It enables us obtain the phase velocity, from which the elastic (or viscoelastic) modulus is deduced, to be evaluated correctly.

Solution of the inverse problem when the dynamic responses of the sample are known

Before carrying out a testing procedure, the experimenter has already chosen an appropriate theory describing the dynamic behavior of the bounded medium (correctly, constituted by the sample in a chosen frequency range). This allows an appropriate mathematical model to be obtained, the dimensions and shape of the sample, as well as the nature of the vibration, being known.

The viscoelastic modulus of the sample material versus the frequency has to be evaluated from measured dynamic responses. That constitutes the solution of the inverse problem, whose mathematical unicity is unfortunately not ensured. Additional problems concerning a numerical solution include numerical instability (depending on the type of differential equations adopted) and the mathematical instability inherent to the type of numerical method adopted (Newton–Raphson's method for example). The experimenter also has to choose an optimization criterion.

Research into a closed form expression of the complex viscoelastic modulus versus frequency1

In the dynamics of a structure, it is necessary to include the constitutive equations of the material in any computer program. The closed form expressions of complex modulus obtained from the experimental responses of the sample have been proposed by many researchers2.

Editors and authors

Editors

All chapters were rewritten by the editors, J.T. Vinh and Y. Chevalier, to bring a coherent approach across the whole book. The two editors were supervisors of research undertaken by all the authors of this book3 in the material engineering laboratory at the Institut Supérieur des Matériaux et de la Construction Mécanique (ISMCM) in France, which was directed by J.T. Vinh from 1965 to 1996; from 1996 to 20094, the laboratory was directed by Y. Chevalier. Each chapter can be considered as a scientific paper with large extracts of the original thesis. Additional sections written by one of the co-ordinators enable us to situate the contribution of the research author among other scientific papers.

Some chapters are extracted from the editors' lecture notes delivered at the aforementioned Institute and at oversea universities when the editors were visiting professors. These complements in mechanics and viscoelasticity might be helpful to the reader.

Contributors

The contributors to the book come from various backgrounds. Many are students in mechanical engineering from engineering colleges in France who prepared written final reports on research undertaken in mechanics during a period of one year or more (equivalent to a Master's degree in English-speaking countries). Others are students who obtained mechanical engineering diplomas, or research fellows who prepared PhD theses or engineering doctoral theses over three years or more.

Presentation of the content of the book

The book is divided into two parts. Part I (Chapters 1 to 8) is devoted to mechanical and electronic instrumentations.

Chapter 1 presents guidelines for the choice of experimental set-ups.

Chapter 2 provides a short review of some of the industrial analyzers available. A sample is often assimilated to a simple model with one degree of freedom mechanical system that is constituted by a spring and dashpot whose coefficients are frequency-dependent and capable of directly describing the material's dynamic behavior. This hypothesis is compatible with the use, at low frequency, of elementary motion equations in which shear and inertia effects of the sample are neglected even for a short sample (see [CHE 10]). Geometric wave dispersion is an important phenomenon in the bounded medium constituted by the sample. If it is ignored, calculations of the viscoelastic coefficient of the material are subjected to large errors when working frequency increases. Coupling vibrations give rise to empirical corrective coefficient in the proposed formulae.

Chapter 3 presents test benches with various mechanical parts. Different sample holders are discussed, in view of adopting the compatible boundary conditions. Ideal boundary conditions transposed into applications and mechanical realizations give rise to additional constraints which have an influence on the effective length (which is different from the measured sample length). A special pseudo-clamping system is proposed so as to avoid, in some special circumstances, sample length correction due to the three-dimensional state of stress in the part of the sample submitted to compressive force. The accuracy of the material complex modulus obtained depends on the choice of the sample holder and the care with which this mechanical part is realized.

Chapter 4 is devoted to electromechanical exciters as well as to piezoelectric and ferroelectric exciters. Various mechanical exciter signals (impulse, sinusoidal, white noise) are discussed. It is shown that, in many cases, we need to make special-sized transducers, or transducers working in unusual frequency ranges, by ourselves.

Chapter 5 deals with transducers in their various forms (displacement, velocity, acceleration, force, etc.). Their choice and correct mounting conditions, and the validity of final results, are reviewed.

Chapter 6 concerns electronic equipment for transducers. Practical considerations about connecting cables are discussed. The equipment devoted to digital signal processing and the useful programs derived from a fast Fourier transform (FFT) are concisely presented. Attention is focused on the transfer function and the coherence function extensively used in dynamic tests.

Chapter 7 is devoted to the Hilbert frequency transform which has recently been applied with success in experimental structural dynamics. The transform enables the detection of the presence of possible hidden non-linearities in the responses of the sample. We are convinced this is very helpful for experimenters.

In Chapter 8, various methods of damping measurements commonly adopted in structural dynamics are presented and discussed. Use of such possibilities extended to material damping evaluation, requires caution, particularly for high damping materials5. Making a distinction between structural mechanical damping and internal material damping avoids confusion. The problem resorts to an inverse problem and the corresponding solution is rarely directly obtained from a structural damping coefficient, except in some cases where explicit formulae are obtained by expansion into series of the eigenvalues versus structural damping coefficients. There are not one but at least two (or possibly more) material damping coefficients, depending on the symmetry of the material itself which conditions the number of coefficients. For composite non-isotropic materials, complex non-diagonal terms of compliance (or stiffness) matrixes require coupled vibrations, of two vibrations of a different nature. Evaluation of damping coefficients becomes difficult if not impossible. The energy partition is not easy to operate.

Part II (Chapters 9 to 15) is devoted to experiments and the experimental interpretation of elastic (or viscoelastic) moduli. After choosing the type of vibration (extensional, bending, and torsional) according to the modulus to be measured, appropriate equations are retained. The next step is to realize the experimental set-up including mechanical and electronic parts.

Just as Part I presents details on instrumentation, Part II presents the aforementioned set-ups concisely. Some suggested set-ups can be adopted in laboratory experiments for students and researchers, to illustrate an advanced course (or research) in elastodynamics. Special experiments can be envisaged to obtain experimental geometrical dispersion curves even at higher modes of vibrations and higher frequency.

Chapter 9 presents two different apparatuses for the torsional vibration of rods: one for lower frequency ranges and the other adapted for higher frequency exploration. The first of these uses a pseudo-clamping of the sample. The symmetry of the sample and the application of excitation of the torque in the middle of the sample permits a moderate clamping force to be adopted for the sample. Length correction is not necessary. Saint-Venant's principle is used and the working frequency is less than 2,000 Hz.

Nugues's theory is used and wave dispersion is discussed taking into account some parameters such as flatness (the ratio of width to thickness) and slenderness (the ratio of width to length).

Onobiono's theory (see [CHE 10]) does not use Saint-Venant's principle but rather Engstrom's theory, extended to anisotropic material. This theory is more appropriate to portray warping phenomenon at higher elastodynamic modes and higher frequency. A special set-up is used for this purpose.

Chapter 10 is devoted to bending tests which are the easiest to realize. As the degree of the differential equations of the motion (Bernoulli–Euler's or Timoshenko–Mindlin's equations) does not exceed 4, systematic use of characteristic functions6 is possible to find the eigenvalue equation of the vibrating rod. Wave dispersion is discussed in relation to some important parameters such as slenderness, flatness and the ratio axial Young's modulus on shear modulus; this last parameter plays an important role in the study of composite materials.

Chapter 11 concerns longitudinal vibration of rods. In classical textbooks devoted to the dynamics of structures, a second order equation of motion is referred to. Experimenters must be careful when choosing such an equation. Wave dispersion is absent from this equation and this assertion might give rise to large errors when evaluating a complex Young's modulus. The validity of this hypothesis must be checked by evaluating the frequency range as a prerequisite condition. Elastodynamic spectrum7 or the curve relative phase velocity versus relative wave number presented in this chapter is appropriate in the determination of the width of frequency range.

When moving from an isotropic material to an anisotropic (composite) material, wave dispersion is much more pronounced and the experimenter's attention is required when choosing appropriate equations of motion.

In Chapter 12, details are given on the conception and fabrication of Le Rolland-Sorin's double pendulum which allows Young's and shear moduli of materials to be evaluated at low frequency. This simple and artful apparatus does not require an external exciter and the only necessary measurements are of the beating period of the pendulums. With a recording of the oscillations of one of the pendulums, it is possible to evaluate the material damping coefficients.

Extension of the use of the double pendulum to measure elastic (or viscoelastic) moduli of anisotropic materials is possible.

Chapter 13 is devoted to measurements of material moduli using rings or hollow cylinders. In many circumstances, a sample is presented with an unusual shape. To fully characterize the material, it is necessary to effect measurements directly on a ring and hollow cylinder successively and also, if necessary, on a straight rod cut-off from a hollow cylinder with a curved cross-section.

In Chapter 14, ultrasonic benches are presented. A water immersion bench is devoted to measurements of two stiffness coefficients. Two direct contact benches permit the evaluation of the remaining material stiffness coefficients. Evaluation of damping coefficients is possible using a logarithmic decrement method.

Chapter 15 is not (yet) devoted to industrial applications. It concerns special devices to evaluate three kinds of waves in turn: torsional, bending and longitudinal waves using long rods and special transducers. These enable experimental evaluation of phase velocities of the three waves in an extremely wide frequency range. Theoretical wave dispersion studies are presented in parallel so as to obtain a higher approximation degree for the motion equations. A variation Hellinger– Reissner's principle with mixed fields (of displacement and stress fields) is used for this purpose. These experiments might be interesting for researchers not only in the domain of metallic materials but also in composite anisotropic materials.

The objective of Chapter 15 is also to show that wave dispersion at high and very high frequencies belongs, in fact, for the moment to academic theoretical studies8. However, in the near future, promising prospective applications can be envisaged, including interesting applications in material characterization and in fracture mechanics.

Yvon Chevalier and Jean Tuong VinhJune 2010

Bibliography

[BED 04] BEDA T. & CHEVALIER Y., “New method for identifying rheological parameter for fractional derivative modeling of viscoelastic behavior”, Mechanics of Time Dependent Materials, Vol. 8, p.105-118, 2004.

[CHE 10] CHEVALIER Y., VINH J.T. (eds.), Mechanics of Viscoelastic Materials and Wave Dispersion, ISTE Ltd, London and John Wiley & Sons, New York, 2010.

[COL 81] COLE K. S. & COLE R. H., J. Chem. Physics, Vol. 9, p.341, 1981.

[KNA 81] KNAUSS W. G. & EMRI I. J., “Non-linear viscoelasticity based on free volume consideration”, Computers & Structures, Vol. 13, p.123-128, 1981.

[MAR 05] MARINOVA S., POPOVA M., CHEVALIER Y., “Friction between warp and weft in viscoelastic behavior of pliable fabric composites”, Sc. Engineering of Composite Materials, vol. 12, N°1-2, p.109-116, 2005.

[NAS 75] NASHIF A. H., JONES D. I. G., and HANDERSON J. P., Vibration Damping, New York, John Wiley & Son, 1975.

[NIN 59] NINOMYA K., FERRY J. D., J. COLLOID . Sci., Vol. 14, p. 417-476, 1959.

[O'DO 95] O'DOWD N. P., KNAUSS W. G., “Time dependent large principal deformation of polymers”, J. Mech. Phys. Solids, Vol. 43, p.771-792, 1995.

[SAD 03] SAAD P., “Modélisation du comportement viscoélastique non linéaire des élastomères autour d'une précharge”, Mécanique & Industries, Vol. 4, p. 133-142, 2003.

[SCHA 99] SCHAPERY R. A., “Non linear viscoelastic and viscoplastic constitutive-equations with growing damage”, International J. Fracture, Vol. 97, p.33-66, 1999.

[VIN 67] VINH J.T., Sur le passage du régime harmonique au régime transitoire, (About the passage of harmonic regime into transient regime) Mémorial de l'Artillerie Française, 3rd Fasc., p.725-776, 1967.

1 In many cases this stage is not necessary for chemists investigating a material. The shape of the complex modulus, particularly the damping coefficients of a material, permits them to distinguish the role of the constituents of the material itself.

2 See references at the end of this introduction.

3 The exception to this is Touratier's doctoral engineering thesis, which was not prepared under the editors' supervision.

4 Since 1998, the Institute has been called the Institut Supérieur de Mécanique de Paris (ISMEP), whilst the laboratory has additional activities and has changed its name to become LISMMA (Laboratory of Engineering of Mechanical Structures and Materials)

5 The material damping coefficient tan δ is equal to or higher than 5x10-2.

6 This function is the linear combination of trigonometric and hyperbolic functions.

7 Represented by the curve relative phase velocity versus the relative wave number.

8 To our knowledge, the devices described (and the experimental higher elastodynamic spectra for composite materials at higher modes) are among the first ever presented.

Acknowledgements

Before interpreting the dynamic responses of rods and plates, measurements taken of these vibrating bounded media seem a priori an easy task when we have at our disposal all the appropriate instruments especially tailored for this task. That was the belief of one of the writers of this book, 40 years ago. Often, however, we are not so lucky.

Over the years, researchers, engineers, students and also technicians have realized that they needed to do a lot of work themselves to improve instruments or invent new ones. The opinions of each researcher and student are invaluable contributions to be taken into account.

As theoretical studies concerning wave dispersions evolve over time, the range of their investigations gradually reaches new unknown frontiers. Our ambition has been to find new and appropriate devices to obtain corresponding experimental results to confirm or to invalidate theoretical results.

In all this research, we have to create appropriate devices for the new measurements ourselves. In this task, the opinions of technicians are as important as those of researchers.

Jean-Baptiste Casimir, Assistant Professor at the ISMEP, is a pioneer in the field of the continuous element method applied to mechanical structures in their various forms (rod, plates, cylinder, etc). He also got the knack of effecting complementary calculations and experiments during the last months of the university year of 2009. He supervised the writing of Chapter 12.

Professor Fei Bin Jun, Vice-President of the University of Aeronautics and Astronautics in Beijing, China, and our old research fellow, kindly sent us his Master of Science report (September 1985) on the Hilbert frequency transform. His pioneering work in this area is presented in Chapter 7.

Professor Maurice Touratier, at the Laboratory of Materials and Structure at the ENSAM High School, Paris, one of our old research fellows, is gratefully acknowledged. The authors whole-heartedly acknowledge Professor Touratier for allowing them to write Chapters 11 and 15 using large extracts from his engineering doctoral thesis1.

Anne Vinh, page layout expert, was a great help to the editors in presenting this book in acceptable form. With great patience, she accepted the ungrateful and heavy burden of supervising corrections to the English presentation of the whole book. Thanks to her, recommendations concerning the practical presentation of the book were finally taken into account.

Elhadi Brahimi's task was to redraw nearly all the innumerable figures including complicated charts with multiple parameters in intricate interlacing networks. He used time and patience, over two whole years, to present drawings in an acceptable form for Chapters 1 to 8.

Finally, last but not least, Cécile Rault, Assistant Editor at ISTE Ltd, is gratefully acknowledged for her valuable advice and useful recommendations to our technical team. Without her help, we would not have reached the deadline we initially promised her.

The English-speaking proof reader who undertook the ungrateful task of correcting numerous inappropriate English words and/or technical vocabulary is gratefully acknowledged. His remarks were a great help for us to improve the final version of our book.

1 His thesis was not prepared under our supervision

PART IMechanical and Electronic Instrumentation

Chapter 1

Guidelines for Choosing the Experimental Set-upa

From an experimental point of view, the elastic and/or viscoelastic characterization of materials is not necessarily achieved simply by using an existing piece of industrial apparatus.

To begin with, the researcher has to choose the experimental set-up, taking the following items into account:

– the type of wave, whether progressive or stationary;

– the measurement technique;

– the numerical method to calculate the elastic (or viscoelastic) modulus or stiffness coefficient.

In this chapter, choice criteria as well as selection guidelines are presented. The following topics will be discussed in turn:

– choice of matrix coefficient(s) (stiffness or compliance matrix) to be evaluated;

– frequency range in which tests are to be conducted;

– shape and dimensions of the sample;

– temperature range to be adopted;

– viscoelastic properties of the material frequency dependence, damping capacity, etc.

– available previsional calculations (for composite materials) which enable the order, or the range, of elastic constants to be obtained.

1.1. Choice of matrix coefficient to be evaluated and type of wave to be adopted

1.1.1. For isotropic materials

The number of elastic constants is reduced to two, chosen from five available elastic constants: Young's modulus, E; shear modulus, G; Poisson's number, ν; volumic dilatation, K; and the stiffness coefficient Ciiii related to an extensional wave. For mechanical applications at low and medium frequency range (f ≤ 10,000 Hz), a compliance matrix [S] is preferred.

Table 1.1. The two classes of tests to be selected when the text material is isotropic

In Table 1.1 the two classes of tests1 permitting the evaluation of a compliance matrix [S] and a stiffness matrix [C] are presented. A bending wave enables the Young's modulus to be obtained, and a torsional wave, the shear modulus. A bending wave is preferred to an extensional wave for many practical reasons2:

– the ease with which measurements are effected;

– a bending wave dispersion is completely portrayed by the fourth order equation of motion (Mindlin–Timoshenko's equation or, with restriction at lower a frequency range, Bernoulli–Euler's equation);

– an extensional wave is the other possibility. However, at medium and higher frequency ranges, a sixth order equation of motion is referred to and consequently it is more difficult to handle the characteristic functions.

The ultrasonic method is easy to carry out. A thick plate sample must be chosen so as to produce, with some care, progressive waves (extensional or shear) in the samples. The wavelength Λ through thickness h satisfies the following inequality:

[1.1]

1.1.2. For anisotropic materials

The number of elastic constants depends on the degree of symmetry of the material. Remember that the number of different constants required for various materials is as follows:

– orthotropic material (wood): 9 constants;

– quasi-transverse (tetragonal) material: 6 constants;

– transverse isotropic material (long fibers regularly distributed in resin matrix): 5 constants;

– quasi-isotropic (cubic) material: 3 constants.

If preliminary information about the material is known (for example the degree of material symmetry [CHE 10]), the samples (number and shape) can be tailored with respect to the symmetry axis of the material. For a rod sample, its axis can be chosen to be coincident or different from the axis of symmetry of the material. For plates in ultrasonic tests, the material axis of symmetry may be collinear (or not) with the plate axis along the thickness, and the propagation direction of waves in any direction is obtained by transducer orientations.

1.1.2.1. Orthotropic material

Figure 1.1 shows three rods (of rectangular or square section) which are fabricated with a rod axis collinear with one material axis. From these three samples, six compliance matrix coefficients can be obtained:

[1.2]

[1.3]

Three remaining non-diagonal coefficients are to be evaluated. For this purpose, three other rod samples are fabricated. These samples are off-axis rods. The angles formed by the rod axis and the reference axis related to the material must be optimized, as in Figure 1.1(b).

The three off-axis samples permit the three non-diagonal compliance coefficients to be evaluated.

Figure 1.1. Orthotropic material samples to be fabricated: a) three rods whose axes are respectively collinear with one of the symmetry axis of the orthotropic material. b) Off-axis rods whose axes are in the planes (1, 2), (2, 3), (3, 1) delimited by the symmetry axes of the material. Rod axes make angles φi ≠ 0, i= (1, 2, 3)

Attention should be focused on the accuracy of the angles φi with which the rod samples in Figure 1.1(b) must be made. As the rod axes do not coincide with the symmetry axes of the material, we have to deal with the change of reference axes for tensor components of the fourth order in formulae giving non-diagonal compliance coefficients, since a weak variation of angle φI gives rise to an important variation of power four (see [CHE 10] Chapter 1, pp. 29-31) of the trigonometric functions. For plates tailored for ultrasonic measurement, we have to deal with a Christoffel's tensor of power 2: the accuracy of the angle φ formed between the normal coordinate system tied up to the plate sample and one of the symmetry axes of the material intervenes in a Christoffel's tensor of power 2, (see [CHE 10] Chapter 10, pp. 517-523) consequently this influence is less “critical” than the aforementioned compliance matrix coefficients.

1.1.2.2. Transverse isotropic material

For artificial composite materials, a transverse isotropic symmetry is usually adopted [CHE 10]. This involves uniaxial long fibers periodically distributed in a resin matrix. Figure 1.2 presents two rod samples whose z axes are, respectively, collinear with symmetry axis 3 and an off-axis rod, whose z axis makes an angle θ with the material plane (1, 2). The third sample is an off-axis whose z axis makes an angle θ ≠ 0 with material axis 3. The two first samples, 1 and 2, enable calculations of the following compliance coefficients:

[1.4a]

[1.4b]

Figure 1.2. Transverse isotropic material with. the material isotropic plane (1, 2) and material symmetry axis 3. Samples 1 and 2 allow four independent compliance coefficients to be obtained. Sample 3 enables calculation of S23

The third sample allows a material non-diagonal coefficient to be obtained and consequently Poisson's numbers ν32 and ν23

[1.5]

The samples used in Figure 1.2 are fragile during fabrication, as well as during measurement manipulation:

- for sample 3, the accuracy of the non-diagonal compliance coefficient strongly depends on the accuracy of the angle θ;

- if the material is strongly viscoelastic (i.e. if the damping capacity is high (tanδ)), the experimenter must be careful when using both the vibration and ultrasonic progressive wave techniques. Results obtained from the two techniques cannot be used concurrently to evaluate the remaining matrix coefficients. Since the working frequencies of waves are not in the same frequency range, the calculations might give rise to large errors.

The second reason to avoid this “mixing” method is that stationary waves require the use of a compliance matrix even though ultrasonic progressive waves concern a stiffness matrix. Matrix inversion is possible if a complete set of experimental stiffness coefficients has already been obtained.

Table 1.2 shows the two classes of testing methods. Details concerning dimensions of samples will be discussed later.

Table 1.2. For transverse isotropic material, five stiffness (or compliance) coefficients should be determined. The two classes of testing methods are used concurrently

1.1.2.3. Orthotropic materials

For ultrasonic testing of massive materials, such as wood4 or artificial three-dimensional composites, three thick plates can be cut (one with an axis collinear with the material's symmetry axis (or trunk axis): a radial plate; one plate at the peripheral; and a tangential plate, perpendicular to material symmetry axis) although difficulty exists in fabricating off-axis plates. For vibration tests on rods, readers should consult Figure 1.1.

1.2. Influence of frequency range

This question is related to the viscoelastic behavior of material. The following remarks might be helpful for experimenters.

1.2.1. The Williams-Landel-Ferry method

It is useful to use the William-Landel-Ferry method to obtain artificial enlargement of the frequency range by using temperature as a variable parameter. The applicability of this method (presented in detail elsewhere: see [CHE 10], Chapter 10) must be valid. It concerns the correspondence (temperature-frequency) principle.

1.2.1.1. Adjustable temperature

The use of a climatic chamber with positive and negative temperature adjustments is appropriate. A gradient of temperature on the rod is, however, to be avoided if time delay is not respected for temperature stabilization during heating or cooling operation.

1.2.1.2. Choice of frequency range

If a narrow frequency range is adopted, the dimensions of the samples should be chosen so as to obtain measurable amplitude of vibration on the sample. Attention should be focused on the first resonance frequencies.

The choice of frequency range is closely related to wave dispersion. Precaution should be taken to evaluate the wave dispersion correctly before evaluating the viscoelastic dispersion.

1.2.1.3. Ultrasonic tests

The working frequency is that of the transducer itself; it should correspond to the frequency (resonance frequency) of the transducer. The temperature is generally the ambient temperature, except in the case where the transducer coupling medium between the transducer and the sample is not a liquid coupling but a special long rod, at the end of which the sample is glued. The sample can be in a special chamber at high temperature, T > 100°C.

1.3. Dimensions and shape of the samples

If a large volume of material is at the experimenter's disposal, plates and rods can be easily fabricated. However, some rod shapes are better suited to testing and calculations. The following practical considerations are useful.

1.3.1. Square section rod for longitudinal wave

If an extensional wave is adopted, a square section is to be preferred to a rectangular section. The reason for this is that the dispersion curve (velocity versus wave number or frequency) is less pronounced for a square section than in the case of a rectangular section, with flatness coefficient b (width)/h (thickness) < 1.

1.3.2. Rod slenderness

Rod slenderness is defined as the ratio h (thickness)/L (length). If the smallest possible slenderness is chosen, a large number of resonance frequencies is obtained. Higher frequencies are thus more easily obtained.

1.3.3. Imposed shape and size

In many cases it is difficult to obtain the shape and size wished for. In tests on bone, for example, the sample may be small in size, with a curved section. There is no possibility of cutting from a larger sample and one cannot manufacture a flat rod sample. One knows that there is a gradient in the elastic properties from the bone center to the free surface. In this case, the ultrasonic technique with special transducers would be the best method to adopt. Curved samples are often imposed on the experimenter (see [CHE 10], Chapter 10).

1.4. Tests at high and low temperature

Elastic and/or viscoelastic properties of materials change with temperature. A temperature controlled room with adjustable temperature between about -70°C to 250°C would be useful. No temperature gradient on the sample should be accepted. For negative low temperatures, a forced preliminary heating ventilation would be useful to avoid ice condensation on the sample.

Special transducers with special connecting cables are necessary for high temperatures.

1.5. Sample holder at high temperature

The sample holder plays an important role. Caution must be taken in bolt and screw systems to maintain the sample firmly without deforming the sample at the contact zones between sample and holder. Clamping systems require special precautions so as to avoid material creep at high temperatures.

1.6. Visual observation inside the ambient room

A glass window is necessary to examine the sample during tests. The window must be able to withstand high temperature.

1.7. Complex moduli of viscoelastic materials and damping capacity measurements

Measurement techniques deserve the special attention of the experimenter. Measurement techniques change drastically depending on the magnitude of the material damping capacity. With a very low damping coefficient of tan δ ≈ 10-3 measurement at ambient atmosphere is subjected to large errors. The first factor to take into account is air damping around the sample, which is of this order of the material's damping tanδ in the interval (10-3-5.10-3). To avoid this disadvantage, a special set-up with a vacuum system is necessary. Damping of the sample holder is also to be taken into account. Precautions concerning measurement techniques will be examined in Chapter 8.

1.8. Previsional calculation of composite materials

Surprisingly, at first sight, this topic is presented as a useful companion to testing. It merits some explanation. In dynamic tests, the experimenter is often confronted with a problem of the magnitude of the first eigenfrequency, the dimensions and size of the sample being known before a test. If the order of elastic moduli is known in advance it will be a great help for the experimenter to choose the frequency range and to possibly discard resonance frequencies due to parasitic oscillations of the sample holder system and the exciter (see [CHE 10], Chapter 1).

The topics presented above constitute only preliminaries which will be expanded in the following chapters.

1.9. Bibliography

[CHE 10] CHEVALIER , Y., and VINH , J.T. (ed.), Mechanics of Viscoelastic Materials and Wave Dispersion, ISTE Ltd, London and John Wiley & Sons, New York, 2010.

a Chapter written by Jean Tuong VINH

1 Indexes are used for anisotropic composite materials and not for isotropic materials.

2 The wave dispersion of an extensional wave requires a sixth order equation of motion to cover the whole frequency range.

3 Figures in subscript are different in tensorial and matrix notations. For shear moduli Gij, subscripts i and j indicate the plane in which shear stress and strain occur.

4 The elastic properties of wood from the center of a trunk to the bark might present a gradient in mechanical properties which constitutes a particular problem to be solved.

Chapter 2

Review of Industrial Analyzers for Material Characterizationa

The issue of instrumentation for material characterization is indeed a very wide subject, covering numerous different areas of concern, such as:

– the mechanical behavior of high polymers;

– the elaboration of artificial composite materials;

– metallic materials for special applications;

– the application of materials in special conditions of temperature and environment;

– the characterization of materials in a large range of temperature and/or frequencies, in view of special applications requiring quantitative information on the viscoelastic behavior of materials;

– correlation between mechanical viscoelastic properties of materials and molecular interpretation, as well as analysis of structure geometry.

There are many books in which the principles of measurement of each of these apparatus are presented in detail, including those by J.D. Ferry [FER 69] and L.E. Nielsen [NIE 74], and I.M. Ward [WAR 71], in a series of chapters, presented a comprehensive discussion of the last item in the list above.

With this book being oriented towards mechanical applications, the mechanical point of view is consequently emphasized. The review of some available industrial analyzers enables the reader to appreciate the mechanical conception of each instrument and understand whether it is suitable for their needs. Often, the mechanical part of the instrument is presented in such a manner that it is hidden or drowned in a complex whole, where the electronic equipment and automatically programmed calculations by computers seem to be the most important part of the analyzer.

Often, the first-time user of this kind of instrument can feel that it is comfortable and easy to use, and almost forgets to ask the main question: is the instrument well adapted for my measurement objectives?

2.1. Rheovibron and its successive versions

2.1.1. Testing of filamentous sample and short rods

Rheovibrons such as DMA 100, 150, 400 (Dynamic Mechanical Analyzers) or VHF 104 allow the measurement of the complex Young's modulus of samples presented as filaments. Figure 2.1 presents the principles of the apparatus. The sample filament is placed in a climatic chamber whose temperature can be adjusted in the range -150°C < T < 300°C.

Figure 2.1. Schematic diagram of a Rheovibron

The forced vibration imposed on the sample is produced by an electromagnetic exciter (on the left of the diagram in Figure 2.1). At the entrance to the oven, a displacement strain gauge transducer measures ∆x imposed to the point i of the filament. The tip k at the right of the sample is connected to a force strain gauge transducer which measures the force Fk. These two transducers give, through two Wheatstone bridges, two electrical tensions Ed and Ef such as:

[2.1a]

[2.1b]

Kd, Ed, KF, EF are proportional constants of the apparatus related to the measurement circuits. From [2.1a] and [2.1b] longitudinal strain and stress are evaluated:

[2.2]

Stars designate complex quantities.

Bringing [2.1] into [2.2], one obtains the complex Young's modulus E*:

[2.3]

The first parenthesis corresponds to a deviation of an electronic instrument, the second and the third parentheses are apparatus constants, and the fourth parenthesis is related to the sample. As the measurement is effected under harmonic regime, there is a phase angle between the two vectors σ* and ε*.

An electric phasemeter permits the measurement of the phase angle between stress and strain. If the electric voltage proportional to σ* is adjusted so that Vσ equals the tension proportional to ε*:

[2.4]

Figure 2.2 shows that the vector joining the two vectors is equal to 2V sin (δE/2).

Figure 2.2. Measurement of the phase angle δ. Vector B is electrically adjusted so that its absolute value equals the absolute value of vector A

Conversion of sin (δ/2) into tan δ is possible:

[2.5]

The apparatus can accommodate a fiber of length 2 < L < 6 mm, with a sample cross-section area s < 0.2 cm2. Since in [2.5] the complex Young's modulus is measured in gain and phase, (E*, δ), the real and imaginary parts of the Young's modulus can be evaluated:

[2.6]

The direct reading of tan δ and the absolute value of E* on the one hand, and the wide temperature range on the other hand, are the main characteristics of this instrument. The measured loss coefficient tan δ is in the range:

[2.7]

2.1.2. Improvement of the Rheovibron: the Rheovibron viscoanalyzer DDV II

The initial conception of the mechanical part of the Rheovibron presents many weak points, detailed as follows:

– the elastic compliance of the mechanical system must be taken into account, including the exciter itself and the connection with the sample, as well as the connection between the sample and the stress gauge;

– at high temperature, samples tend to yield between grips. Correction is necessary;

– Massa [MAS 73] suggested mechanical improvements for the inertia of the mechanical system, and proposed correction factors for varying temperature and frequency as well as for the dimensions of the samples themselves, and even for the nature of the material.

This raises the problem of direct connection of the sample via sample holders and the mechanical environment, which is difficult to introduce into the equation given the complex Young's modulus, and complex additional dynamic systems (see Figure 2.3). These last set of mechanical components are made up of spring, dashpot (which are frequency dependent and temperature dependent) and inertia of exciter and transducer.

Figure 2.3. Having an exciter and a mechanical sample holder introduces two sets of components on each side of the sample, constituted by spring ki, dashpot ηi and mass mi

Modification of the sample holders has been proposed to adapt the instrument for bending tests. Erhard [ERH 70] suggested new holders for shearing tests. Compression tests and bending tests were proposed by Murayama [MUR 67] for measurements on anisotropic materials.

A servo-hydraulic actuator was introduced to increase the dynamic load capacity in magnitude as well as in a lower frequency range (f ≅ 0.1 to 5Hz), and a piezoelectric transducer was introduced for force measurements.

2.1.3. Automated and improved version of Rheovibron by Princeton Applied Research Model 129 A

This model, an automated Rheovibron by Princeton Applied Research, introduces two phase locks in the amplifier and data logging system.

2.2. Dynamic mechanical analyzer DMA 01dB – Metravib and VHF 104 Metravib analyzer

This French apparatus is initially adapted for short samples working in dynamic compression tests. Figure 2.4 shows a material presented as a short cylinder between two masses, m1 and m2. The mechanical holder and sample work as a two degrees of freedom system. The frequency response of the system presents a resonance and anti-resonance. The useful frequency part of the system is between these two extremes. The complex stiffness k* of the sample, magnitude |k*(ω)| and the phase (damping) angle δE are evaluated:

Figure 2.4. A 01dB Metravib instrument: (a) schematic diagram of loading system and displacement measurements x1 and x2; (b) gain and phase response of the system, presentingresonance and anti-resonance. The working frequency interval is between the two zones. |Z| is the transfer function x2*/ x1*and ϕ is the phase angle of the whole system (the sampleand additional masses)

2.2.1. Comments

To avoid buckling, a sample working in compression must be short, which raises the question of three-dimensional stress states in the sample. The influence of shear stress as well as the influence of friction at both ends of the sample between the two masses m1, m2 and the sample must be taken into account.

Let us mention that the 01dB Metravib manufacturer proposes a large range of analyzers with a complete set of sample holders which allows compression, tension, bending tests as well as torsion and shearing tests with a variety of shapes and dimensions of samples. In the domain of mechanical analyzers with VHF apparatus, the frequency interval is extended from 100 Hz to 10 kHz. Depending on the adopted version, the ambient temperature of the sample can be chosen from between -50 °C and +110 °C.

2.3. Bruel and Kjaer complex modulus apparatus (Oberst Apparatus)

Bruel and Kjaer, a Danish company, specializes in electronic dynamic measurements (frequency spectrometers, frequency oscillators and level recorders). This company proposes a mechanical system in which the sample is presented as a vibrating reed working in forced flexural vibrations, also referred to as an “Oberst apparatus”.

The sample is clamped at one end (Figure 2.5). A force transducer also working as a force exciter is located at the lower end. The displacement transducer is a contactless one, located at a point between the clamping ends.

Measurements of the flexural Young's modulus | E* | and damping tan δEare effected at and around resonance frequencies of the sample:

[2.8]